2. Bohr Model and QT of Hydrogen Atom

7/30/2019 2. Bohr Model and QT of Hydrogen Atom

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Bohr model of atom

How to see

Early modelsAtomic spectra

The Bohr model

Correspondence principleDemerits…





Early models of atom

Plum pudding model of 1890s

Rutherford’s experiment in 1911



Atomic spectra



Atomic spectra: the key

Planetary model (Rutherford) of atom

proton

electron

An atomic electron

should, classically, spiral

rapidly into the nucleus

as it radiates energy due

to its acceleration



The Bohr atom

Electron waves & stationary states

de Broglie wavelength of e : hmv

Centripetal Electric F F 2 2

2

0

1

4

mv e

r r

04

ev

mr

04Orbital e wavelength:

r h

c m

This corresponds to the circumference of e orbit: 2 r

An e can circle a nucleus only if its orbit contains

an integral number of de Broglie wavelengths

Condition for orbit stability: 2 1, 2,3,nn r n



Stationary states



Bohr orbits

Condition for orbit stability: 2 1, 2,3,nn r n

042n

n

r nhr

e m

2 20

2Orbital radii in Bohr atom: 1,2,3,...n

n hr n

me

11 2

0 1 0Bohr radius: 5.292 10 ; na r m r n a

Angular momentum quantization (alternate approach)

h

mv 2

n

n r

2( )( / )mr v r mvr L I n2

hn



Energy levels

2 2

04

mv e E KE PE

r r

04

ev

mr

2

04n

n

e E

r

4

12 2 2 2

0

1 1,2,3,...8

n E me E nh n n







12 2

1 1 1:l u

E Hydrogen spectrumhc n n



Correspondence principle

The greater the quantum number, the closer

quantum physics approaches classical physics

At very high ‘n’ we have more dense

levels which are more like continuum



Bohr atom model: Demerits

Applicable only to hydrogen and other one-

electron ions such as He+

and Li2+

Cannot explain why some lines are more

intense than others

Cannot explain why many lines consist of several separate lines whose wavelengths

differ very slightly

No light on how individual atoms interact Quantum mechanics was developed

(1925,1926:Schrodinger, Heisenberg, Born,

Dirac & others) to overcome these shortfalls



Quantum theory of

hydrogen atom

• Quantum mechanics: recap

• Schrodinger’s eqn. for Hydrogen atom • Separation of variables

• Quantum numbers

• Electron probability density• Selection rules

• Zeeman effect

Q t h i



Quantum mechanics: recap

Explores probabilities instead of asserting

Eg. Hydrogen atom: r g.s. from Bohr theory = 5.3×

10-11

m QMMost probable r g.s.= 5.3×10-11 m

Wave function

Ψ itself has no physical interpretation

|Ψ|2 probability of finding the body (+ve, real quantity)

2 Normalization: 1dV

2

1 21

2Probability:

x

x x x

P dx

2 2

2 2 2

1Wave equation: (same sense of II law)

y y

x v t

2 2

2

Schrodinger equation: (1D)2

i U t m x

S h di ’ ti



Schrodinger’s equation

2 2 2

2 2 2 2

2( ) 0 (3D)

m E U

x y z

2

0

Electric potential energy:4

eU

r

In spherical polar coordinates, the Schrodinger’s equation becomes

2 2

2 2 2 2

2 2

0

sin sin sin

2 sin0

4

r r r

mr e E

r

S ti f i bl



Separation of variablesHydrogen atom wave fn.: ( , , ) ( ) ( ) ( )r R r

Simply, if R R dRr r dr

2 2

2 2Similarly ,

d d R R

d d

Substituting above in Schrodinger’s eqn. and rearranging,

2 2

2 2 2 2

2 2

0

sinsin sin

2 sin 14

d dR d d r

dr dr d d

mr e d E r d

22

2

1l

d m

d

if ( ) ( ), then ( ) ( ) . f x g y f x g y const



Substituting for ml and rearranging, yields

22 22

2 20

1 2 1sin

4 sin sin

l md dR mr e d d r E

R dr dr r d d

=l(l + 1) (Again we have different variables on both sides)

22

2Equation for : 0l

d m

d

2

2

1Equation for : sin ( 1) 0

sin sin

l md d l l

d d

22

2 2 2

0

1 2 ( 1)Equation for : 0

4

d dR m e l l R r E R

r dr dr r r

Q t b



Quantum numbers

The solution of equation for is given by ( ) l im Ae

Exploiting the symmetry that and + 2 identify the same

Plane, we have,

( ) ( ) ( 2 )l l im im Ae Ae 0, 1, 2, 3,...m

The differential equation for has a solution provided:is an integer and 0, 1, 2,...,l l l m m l

The final solution of radial part yields,

4

12 2 2 2 2

0

1; 1,2,3,... ; 1

32n

E me E n n l

n n

0,1,2,..., ( 1)l n

Thus the principal (n), orbital (l ) and magnetic (m)

quantum numbers are defined

O bit l t b



Orbital quantum number 2

2

2 2 2

0

1 2 ( 1)Equation for : 0

4

d dR m e l l R r E R

r dr dr r r

E includes electron’s orbital kinetic energy also !!

radial orbital E KE KE U 2

04

radial orbital

e KE KE

r

22

2 2 2

1 2 ( 1)0

2radial orbital

d dR m l l r KE KE R

r dr dr mr

If R(r) has to be an exclusive function of r, 2

2

( 1)

2

orbital

l l KE

mr

21

2orbital orbital KE mv

2

22

orbital

L L mv r

mr

2 2

2 2

( 1)

2 2

L l l

mr mr

Electron angular momentum ( 1) L l l

M ti t b



Magnetic quantum number e- revolving around the nucleus minute current loop

Has a magnetic field like that of magnetic dipole

ml specifies the direction of L by

determining the componentof L in the field direction.

Interacts with external magnetic field B

Space quantization

0, 1, 2,..., z l l L m m l

U t i t i i l & L



Uncertainty principle & Lz

Why only L z is quantized?

L can never point any specific direction

but in cone where L z =ml

If not the uncertainty principle will be

violated

If L were in z direction, e- is confined to

xy plane and hence z = 0, p z

L precesses constantly about z -axis

El t b bilit d it



Electron probability density

No definite orbits

QM i f t



QM view of atoms

The orbitals



The orbitals

Selection rules



Selection rules

*Allowed transitions: 0 , ,l l n l m nlmu u x y z

Transitions not obeying above condition are forbidden transitions

Selection rules: 1l 0, 1l m

( )l n l m

( )l nlm

Interaction with magnetic field



Interaction with magnetic fieldThe torque t on a magnetic dipole in a

magnetic field of flux density B is

sin { B r F t

Potential energy 0 when / 2.mU

/2

For other orientations mU d

t

/2

sin B d

cos B

IA 2ef r 2v r r f

22 L mvr mfr

Electron magnetic moment

2

e L

m

Gyromagnetic ratio



cosmU B

2

e

Lm

cos2

m

eU LB

m

( 1) L l l

cos( 1)

l m

l l

2m l

eU m B

m

Bohr magneton:2

B

e

m

In a magnetic field, the energy of a

particular atomic state depends on alsol m

m l BU m B

Zeeman effect



Zeeman effect

In a magnetic field, the energy of a

particular atomic state depends on alsol m

a level with unique ' ' splits into

different levels having different ' 'l

n

m

1 0 0

2 0

3 0 0

4 Normal Zeeman

effect

4

B

B

B ev v v Bh m

v v

B e

v v v Bh m

0, 1l m

m l BU m B

0 m E U E v

h h

0 l Bv v m B



Anomalous Zeeman effect



Anomalous Zeeman effect

The previous QM treatment could not explain both anomalous Zeeman effect and

fine structure

Two Dutch graduate students (Samuel Goudsmit & George Uhlenbeck) proposed in

1925 that

Every e- has an intrinsic angular momentum, called spin, whose magnitude is the

same for all electrons. Associated with this angular momentum is a magnetic

moment

Electron spin



Electron spin1 1 3

Spin angular momentum: ( 1) 12 2 2

S s s

Classical model of a spinning electron. This model gives an

incorrect magnitude for the magnetic moment, incorrect

quantum numbers, and too many degrees of freedom. Spinarises from relativistic dynamics.

1

2 z s

S m

Spin magnetic moment: S

eS

m

2Sz B

e

m

Stern Gerlach experiment



Stern-Gerlach experiment

Cause for deflection: cos z S

dB F

dz

Magnetic moment of silver atom is due to one electron

First proof of space quantization

Spin orbit coupling



Spin-orbit couplingcos ,mU B

cos Sz B

m BU B



Vector atom model



Vector atom model

Total angular momentum: J L S

1( 1) ,

2 J j j j l s l

, , 1,..., 1, z j j j m m j j j j

Precession of L S & J



Precession of L, S & J

J is also space quantized

LS Coupling



LS Coupling How to couple angular momenta in many electron atoms

, ,i i

i i L L S S J L S

H line



Ha

line

Selection rule

1l

For many e atoms

11

0

L J

S

More complications exist• Relativistic effects

• Vacuum fluctuations, etc.

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2. Bohr Model and QT of Hydrogen Atom